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New J. Phys. 21 (2019) 023005
https://doi.org/10.1088/1367-2630/aaf839
PAPER
On the existence of superradiant excitonic states in
microtubules
GLCelardo1,MAngeli2,3, T JACraddock4,5,9 and PKurian6,7,8,9
1 Benemérita Universidad Autónoma de Puebla, Apartado Postal
J-48, Instituto de Física, 72570,Mexico2 International School for
Advanced Studies (SISSA), Via Bonomea 265, I-34136Trieste, Italy3
Dipartimento diMatematica e Fisica and Interdisciplinary
Laboratories for AdvancedMaterials Physics, Università Cattolica
del Sacro
Cuore, viaMusei 41, I-25121 Brescia, Italy4 Departments of
Psychology andNeuroscience, Computer Science, andClinical
Immunology, Nova SoutheasternUniversity, Fort
Lauderdale FL 33314,United States of America5 Clinical Systems
BiologyGroup, Institute forNeuro-ImmuneMedicine, Fort Lauderdale FL
33314,United States of America6 QuantumBiology
Laboratory,HowardUniversity,WashingtonDC20059,United States of
America7 Center for Computational Biology andBioinformatics,
HowardUniversity College ofMedicine,WashingtonDC20059,United States
of
America8 Department of Physics andAstronomy,University of Iowa,
IowaCity IA 52242,United States of America9 Authors towhomany
correspondence should be addressed.
E-mail: [email protected],[email protected],
[email protected] [email protected]
Keywords: quantumbiology, quantum transport in disordered
systems, open quantum systems, energy transfer
AbstractMicrotubules are biological protein polymers with
critical and diverse functions. Their structuresshare some
similarities with photosynthetic antenna complexes, particularly in
the orderedarrangement of photoactivemolecules with large
transition dipolemoments. As the role ofphotoexcitations
inmicrotubules remains an open question, herewe analyze
tryptophanmolecules,the amino acid building block ofmicrotubules
with the largest transition dipole strength. By takingtheir
positions and dipole orientations from realisticmodels capable of
reproducing tubulinexperimental spectra, and using
aHamiltonianwidely employed in quantumoptics to describe
light–matter interactions, we show that suchmolecules arranged in
their nativemicrotubule configurationexhibit a superradiant lowest
exciton state, which represents an excitation fully extended on
thechromophore lattice.We also show that such a superradiant state
emerges due to supertransfercoupling between the lowest exciton
states of smaller blocks of themicrotubule. In the dynamics wefind
that the spreading of excitation is ballistic in the absence of
external sources of disorder andstrongly dependent on initial
conditions. The velocity of photoexcitation spreading is shown to
beenhanced by the supertransfer effect with respect to the velocity
onewould expect from the strength ofthe nearest-neighbor coupling
between tryptophanmolecules in themicrotubule. Finally,
suchstructures are shown to have an enhanced robustness to static
disorder when compared to geometriesthat include only short-range
interactions. These cooperative effects (superradiance and
super-transfer)may induce ultra-efficient photoexcitation
absorption and could enhance excitonic energytransfer
inmicrotubules over long distances under physiological
conditions.
1. Introduction
From the suggestion [1, 2] that coherent wave behaviormight be
implicated in excitonic transport for naturalphotosynthetic systems
under ambient conditions, amplemotivation has since arisen to
investigate therelevance of quantummechanical behavior in diverse
biological networks of photoactivemolecules. Forinstance, in
photosynthetic systems, great attention has been devoted to the
antenna complexes. Such complexesaremade of a network of
chlorophyllmolecules (photoactive in the visible range), which are
able to absorbsunlight and transport the excitation to a
specificmolecular aggregate (the reaction center). The reaction
centeris where charge separation occurs, in order to trigger the
ensuing steps required for carbon fixation [3].
OPEN ACCESS
RECEIVED
20August 2018
REVISED
4December 2018
ACCEPTED FOR PUBLICATION
12December 2018
PUBLISHED
5 February 2019
Original content from thisworkmay be used underthe terms of the
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Any further distribution ofthis workmustmaintainattribution to
theauthor(s) and the title ofthework, journal citationandDOI.
© 2019TheAuthor(s). Published by IOPPublishing Ltd on behalf of
the Institute of Physics andDeutsche PhysikalischeGesellschaft
https://doi.org/10.1088/1367-2630/aaf839https://orcid.org/0000-0002-4160-6434https://orcid.org/0000-0002-4160-6434mailto:[email protected]:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aaf839&domain=pdf&date_stamp=2019-02-05https://crossmark.crossref.org/dialog/?doi=10.1088/1367-2630/aaf839&domain=pdf&date_stamp=2019-02-05http://creativecommons.org/licenses/by/3.0http://creativecommons.org/licenses/by/3.0http://creativecommons.org/licenses/by/3.0
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Some of the dominant coherent effects which are thought to be
responsible for the high efficiency of naturalphotosynthetic
complexes are induced by the delocalization of the excitation
overmanymolecules10. Suchdelocalized excitonic states can lead to
cooperative effects, such as superabsorption and supertransfer [3,
4], andthey can be useful in both natural and engineered
light-harvesting complexes [5–20]. Specifically,
delocalizedexcitonic states can have amuch larger dipole strength
than that of the constituent chromophores, and suchgiant transient
dipoles [21–23] can strongly couple to the electromagnetic field.
Thus, these states are able tosuperabsorb light, i.e. they are able
to absorb light at a rate which ismuch larger than the
single-moleculeabsorbing rate [23]. Indeed, the absorption rate of
delocalized excitonic states can increase with the number
ofmolecules over which the excitation is delocalized [22, 23].
Supertransfer is described in a similar way, withrespect tomovement
of the excitation to an externalmolecular aggregate or between
different parts of the samesystem [4]. Specifically, an excitonic
state delocalized onNmolecules of onemolecular aggregate can couple
withan excitonic state delocalized onMmolecules of a second
aggregate with a coupling amplitudewhich is NMtimes larger than the
coupling amplitude between singlemolecules belonging to different
aggregates. Suchsupertransfer coupling is able to enhance the
velocity of spreading of photoexcitations, and it has been shown
tohave an important role in natural photosynthetic systems
[24].
The role of coherent energy transfer has been investigated not
only in photosynthetic complexes but also inother important
biomolecular polymers, such as in cytoskeletalmicrotubules [25, 26]
and inDNA [27]. In thispaperwewill focus on the role of
photoexcitations inmicrotubules, which are essential biomolecular
structuresthat havemultiple roles in the functionality of cells.
Indeed,microtubules are present in every eukaryotic cell toprovide
structural integrity to the cytoskeletalmatrix, and they are
thought to be involved inmany other cellularfunctions,
includingmotor trafficking, cellular transport,mitotic division,
and cellular signaling in neurons.Interestingly,microtubules share
some structural similarities with photosynthetic antenna complexes,
such asthe cylindrical arrangement of chlorophyllmolecules in
phycobilisome antennas [28] or in green sulphurbacteria [29], where
cylindersmade ofmore than 105 chlorophyllmolecules can efficiently
harvest sunlight forenergy storage in the formof sugar. Note
thatwhile chlorophyllmolecules are active in the visible range
ofelectromagnetic radiation,microtubules possess an architecture of
chromophoricmolecules (i.e. aromaticamino acids like
tryptophan)which are photoactive in the ultraviolet (UV) range.
It remains an open questionwhethermicrotubules have any role in
transporting cellular photoexcitations.Intriguingly, several groups
have studied and experimentally confirmed the presence of veryweak
endogenousphoton emissionswithin the cell across theUV, visible,
and IR spectra [30–34]. It has also been suggested
thatmicrotubulesmay play a role in cellular orientation and
‘vision’ via the centrosome complex [35], and veryrecently two of
us have proposed neuronal signaling pathways inmicrotubules via
coherent excitonictransport [26].
Eachmammalianmicrotubule is composed of 13 protofilaments, which
form a helical-cylindricalarrangement of tubulin subunit protein
dimers, as infigure 1. The tubulin subunit proteins possess a
uniquenetwork of chromophores, namely different amino acids, which
can form excited state transition dipoles in thepresence of
photons. The geometry and dipolemoments of these amino acids, which
are termed aromatic owingto their largely delocalizedπ electrons,
are similar to those of photosynthetic constituents, indicating
thattubulinmay support coherent energy transfer. Aswith
chlorophyllmolecules, it is possible to associate to eacharomatic
amino acid a transition dipole that determines its coupling to
othermolecules andwith theelectromagnetic field.
Themain questionwe address in this paper is whether the
arrangement of photoactivemolecules in themicrotubule structure can
support extended excitonic states with a giant dipole strength, at
least in the absenceof environmental disorder. Such extended
states, if robust to noise, can also support efficient transport
ofphotoexcitations, which could have a biological role
inmicrotubule signaling between cells and across the brain[26]. To
answer this question, we consider first a quantumdescription of the
network of tryptophanmolecules,which are the greatest contributor
to photoabsorption inmicrotubules in theUV range. Indeed,
tryptophan isthe aromatic amino acidwith the largest transition
dipole (6.0 debye), comparable with that of chlorophyllmolecules.We
proceed bymodeling these tryptophans as two-level systems, as is
usually done in photosyntheticantenna complexes. This is, to our
knowledge, the first analysis of excitonic states distributed
across tryptophanchromophore lattices in large-scale,
realisticmodels ofmicrotubules.
The interaction between the transition dipoles of the
photoactivemolecules is in general very complicated,with the common
coupling to the electromagnetic fieldmore nuanced than simple
dipole–dipole interactions,which are an effective description of a
chromophoric network only in the small-size limit (where the system
size
10There is substantial debate among experimentalists regarding
the role of delocalization in excitonic transport. In part this is
due to a
multiplicity of definitions for ‘coherence.’ In this paper
wewill not discuss coherence in the context of oscillatory beating
patterns observedin the exciton state overmultiple chromophores [1,
2]. Instead, we are interested in the collective response of a
network of chromophores,excited by light whosewavelength is of the
same order as the network size.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
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ismuch smaller then thewavelength) [36]. However, for large
aggregates one needs to go beyond the simpledipole–dipole
interactions used in small aggregates. Here we consider an
effective non-HermitianHamiltonianinteraction commonly used in the
literature to study the coupling inmolecular aggregates [22]. This
non-Hermitian description also allows the possibilities of donating
the excitation back to the electromagnetic fieldthrough photon
emission or of transferring excitation coherently between
chromophores.Moreover, in thesmall-system-size limit it reduces to
a dipole–dipole interaction.
The imaginary part of the complex eigenvalues E i 2 = - G of
such a non-HermitianHamiltoniandetermines the strength of the
coupling of the excitonic states with the electromagnetic field and
is connectedwith the dipole strength of the eigenstates of the
system.While the coupling of a single aromaticmolecule can
becharacterized by its decay rate γ,Γ determines the coupling of
extended excitonic states with the electromagneticfield.
Superradiant states are characterized byΓ>γ, while subradiant
states are characterized byΓλ. Note thatÿ/Γ is the lifetime of the
excitonic eigenstate, so that larger values ofΓ govern
fasterexcitation decays. Since the process is symmetric under time
reversal, fast decaying states are also fast absorbingstates. The
advantage of this formalism,with respect to the simple
dipole–dipole interaction commonly used inthe literature, is that
it allows us to consider system sizes that are even larger than
thewavelength of the absorbedlight. This property becomes
particularly important for large biopolymeric structures
likemicrotubules whoselength is generally several orders
ofmagnitude larger than thewavelength associatedwith
themoleculartransitions (λ=280nm). Herewe considermicrotube lengths
up to∼3λ.
We use data on the positions, dipole orientations, and
excitation energies of tryptophanmolecules, whichhave been obtained
bymolecular dynamics simulations and quantum chemistry calculations
[25, 26]. Thesedata have been shown to reproducewell the
absorption, circular dichroism, and linear dichroism spectra
ofsingle tubulin dimers [25, 26].
Our analysis shows that as the number of tubulin subunits
considered grows, a superradiant state forms inthe lowest exciton
state of the system. This is exactly what happens inmany
photosynthetic antenna complexes,such as in green sulfur bacteria
cylindrical antennas [19, 20, 37] and in self-assembledmolecular
nanotubes[38–43]. Superradiant states favor the absorption of
photons by themicrotubule.Moreover, since thesuperradiant lowest
exciton state represents an extended (delocalized) excitonic state
of the order of themicrotubule length, such superradiant states
could serve as a support for efficient transport of
photoexcitation.
Figure 1.Panel (a): tubulin dimer andmicrotubule segment. Left:
solvent-excluded tubulin heterodimer surface withα-tubulinmonomer
in light grey, andβ-tubulinmonomer in dark grey (scale bar∼5nm).
Right: section ofmicrotubule B-lattice structure atthree angles
showing left-handed helical symmetry and protofilament (bottom,
outlined in black box) (scale bar∼25nm). Inmicrotubules, tubulin
dimers stack end-to-end to form the protofilaments, 13 of which
join side-by-side with longitudinal offset andwrap around to form a
tubewith such helical symmetry. Panel (b): arrangement of
tryptophan amino acids inmicrotubule segment atthree angles with
transition dipole directions. Left: a single spiral of tubulin
dimers (light greyα-tubulin, dark greyβ-tubulin) frommicrotubule
structure showing tryptophan amino acids (blue sticks) and
transition dipole directions (red arrows). Right: tryptophanamino
acids and transition dipole directions only (scale bar∼25nm).
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New J. Phys. 21 (2019) 023005 GLCelardo et al
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In the next sectionwe develop themathematicalmachinery for our
physicalmodel, using an effectiveHamiltonian that has beenwidely
used to describe a single photoexcitation interactingwithin a
network oftransient dipoles. In section 3we display several results
demonstrating the existence of a superradiant excitonicstate
extended overmore than 104 tryptophanmolecules of themicrotubule.We
also show that the superradiantlowest exciton state emerges from
the supertransfer coupling between the lowest exciton states of
smallersegments inside themicrotubule. Section 4 shows initial
studies of the exciton dynamics, showing thatcooperativity can
enhance the coupling between different parts of themicrotubule
through supertransfer.Section 5 demonstrates the robustness of the
superradiant lowest exciton state to disorder, andwe close
insection 6with some conclusions and our future outlook.
2. Themodel
Microtubules are cylindrical-helical structuresmade of
essentially two closely related proteins,α- andβ-tubulin. They are
arranged as infigure 1 to form a left-handed helical tube of
protofilament strands. In eachα–βdimer there aremany
aromaticmolecules: eight tryptophans (Trps)whose transition dipoles
are arranged as infigure 1 (see appendix A for complete
description). Their peak excitation energy is∼280 nm, and
themagnitudeof their dipolemoment is 6.0 debye. There also exist
other aromatics, including tyrosine, phenylalanine, andhistidine,
withmuch smaller dipolemoments. For example, tyrosine (themolecule
with the second largestdipole) has a dipolemoment of only 1.2
debye. For the purposes of this initial analysis, we limit our
attention tothe Trps only because of their relatively large
transition dipoles. The position and orientation of the
dipolemoments of Trpmolecules have been obtained frommolecular
dynamics and quantum chemistry calculations,and they reproduce
closely the linear and circular dichroism spectra of tubulin for
the Trp-only case [25, 26].
The interaction of a network of dipoles with the electromagnetic
field is well described by the effectiveHamiltonian [22, 44,
45]
H H Gi
2, 1eff 0= + D - ( )
whereH0 represents the sumof the excitation energies of
eachmolecule, andΔ andG represent the couplingmatrices (elements
listed below in equations (3)–(5)) between themolecules induced by
the interactionwith theelectromagnetic field. Note that such an
effectiveHamiltonian has beenwidely used tomodel
light–matterinteractions in the approximation of a single
excitation. The site energies are all identical, so that we
have
H n n n n n m
G G n n G n m
, ,
. 2
n
N
n
N
nnn m
N
nm
n
N
nnn m
N
nm
01
0å å å
å å
w= ñá D = D ñá + D ñá
= ñá + ñá
= ¹
¹
∣ ∣ ∣ ∣ ∣ ∣
∣ ∣ ∣ ∣ ( )
Thewavenumber associatedwith each site energy is k n cr0 0w≔ ,
where c is the speed of light and nr r rm=is the refractive
index.Most naturalmaterials are non-magnetic at optical frequencies
(relative permeabilityμr≈1) sowe can assume that nr r~ . The real
and imaginary parts of the intermolecular coupling are givenon the
diagonal, respectively, by
G k0,4
3, 3nn nn
r
2
03
m
gD = = ≕ ( )
with m m=∣ ∣being themagnitude of the transition dipole of a
single tryptophan and r being the relative
permittivity, and on the off-diagonal, respectively, by [9, 22,
44]
k r
k r
k r
k r
k r
k r
k r
k r
k r
k r
k r
k rr r
3
4
cos sin cos
cos3
sin3
cos, 4
nmnm
nm
nm
nm
nm
nmn m
nm
nm
nm
nm
nm
nmn nm m nm
0
0
0
02
0
03
0
0
0
02
0
03
gm m
m m
D = - + +
- - + +
⎡⎣⎢⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
( )( )
( )( )
( )( )
ˆ · ˆ
( )( )
( )( )
( )( )
( ˆ · ˆ )( ˆ · ˆ ) ( )
Gk r
k r
k r
k r
k r
k r
k r
k r
k r
k r
k r
k rr r
3
2
sin cos sin
sin3
cos3
sin, 5
nmnm
nm
nm
nm
nm
nmn m
nm
nm
nm
nm
nm
nmn nm m nm
0
0
0
02
0
03
0
0
0
02
0
03
gm m
m m
= + -
- + -
⎡⎣⎢⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
( )( )
( )( )
( )( )
ˆ · ˆ
( )( )
( )( )
( )( )
( ˆ · ˆ )( ˆ · ˆ ) ( )
where n nm m mˆ ≔ is the unit dipolemoment of the nth site and r
r rnm nm nm
ˆ ≔ is the unit vector joining thenth and themth sites. In the
followingwe assume n1r r = = , corresponding to their values in
vacuum/air.
4
New J. Phys. 21 (2019) 023005 GLCelardo et al
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The actual dielectric constant and refractive index of tubulin
is currently debated [26, 34], but using the tubulindielectric
instead of air would increase the imaginary part of the coupling by
2 8.41r , dependingon the value chosen, whichwould proportionally
decrease the lifetimes of the excitonic eigenstates. The real
partof the off-diagonal couplingwould also be increased by the same
factor, augmenting the dipole strength of theexcitonic state.
The eigenvalues of thisHamiltonian are complex, endowing the
eigenstates with a finite lifetime due to theircoupling to the
external environment. The imaginary part of an eigenvalue is
directly linked to the decay rateΓ ofthe eigenstate. Thus for each
eigenmode of the system,Γ represents its coupling to the
electromagnetic field.ForΓ>γwehave excitonic states which are
coupled to the fieldmore strongly than the single
constituentmolecule, representing superradiant states. On the other
hand, the states for whichΓ
-
It should be noted that for small systems k r 1nm0 ( ) the
coupling terms in theHamiltonian (1) become
G
r r
r
,
3. 7
nm n m
nmn m n nm m nm
nm3
gm mm m m m
D-
ˆ · ˆ
· ( · ˆ )( · ˆ )( )
Herewe have neglected terms that go as 1/rnm because they are
dominated by r1 nm3 contributions. In this limit,
the real partΔnm represents a dipole–dipole interaction
energywith n mm m m= = ∣ ∣ ∣ ∣and the radiative decay
width k43
203g m= . Recall that this familiar dipole–dipole couplingwhich
is used to describe the interactions
between the transition dipoles of photoactivemolecules cannot be
used in our case, since this approximation isvalid only when the
size of the system L ismuch smaller then thewavelengthλ
associatedwith the transientdipole. In our analysis we
considermicrotubule lengths which are larger thanλ. Thus in our
case it ismandatoryto go beyond the dipole–dipole approximation
(see discussion in appendix B).
3. Superradiance in the lowest exciton state
Wehave diagonalized the full radiativeHamiltonian given in
equation (1) formicrotubule segments of differentsizes, up to
amicrotubulemore than 800nm long and comprised of 100 spirals,
including a total of 10 400Trpmolecules, so that L/λ≈3. For each
eigenstate and complex eigenvalue E i 2 = - G , we plot the
decaywidthΓ of each state normalized to the single dipole
decaywidth 2.73 10 cm3 1g = ´ - - .
Infigure 2we showhow cooperativity (superradiance) emerges as we
increase the number of spirals in themicrotubule segment (where
each spiral contains 104 Trpmolecules). For one spiral (figure 2),
there is a verydisordered distribution of the decaywidths, but as
we increase the number of spirals a superradiant lowestexciton
state clearly emerges with a decaywidthΓ>γ that increases as we
increase the length of the
Figure 2.Panels (a)–(f): Normalized decay widthsΓ/γ of the
excitonic eigenstates are plotted versus their energies
formicrotubulesegments of different lengths (number of spirals).
Note that each spiral contains 104 tryptophanmolecules and extends
about 9nmin the longitudinal direction. In panel (c) themaximum
length of themicrotubule segments considered in this paper is
shown,comprised of 100 spirals and 10 400 tryptophanmolecules. In
panel (e) amicrotubule of the same length (100 spirals) is
shown;the positions of the tryptophans are the same as in panel
(c), but the orientations of their dipoles are randomized. In panel
(f) amicrotubule of 100 spirals is shown; the positions of the
tryptophans are the same as in panel (c), but the orientations of
their dipolesare randomized in only one spiral and then repeated in
all the other spirals. Finally in panel (d) the location in the
energy spectrumofthe superradiant state is shown as a function of
the number of spirals. The superradiant state coincides with the
lowest exciton state(state 1) for allmicrotubule segments of length
greater than 12 spirals.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
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microtubule segment. Infigure 2 the normalized decaywidthsΓ/γ
versus energies of the eigenstates of themicrotubule comprised of
100 spirals are shown. As one can see,most of the decaywidth is
concentrated in thelowest exciton state. The lowest-energy
superradiant state infigure 2 corresponds to∼600 times the
single-molecule decay rate. Infigure 2 the location in the energy
spectrumof the largest superradiant state is shown as afunction of
the number of spirals. The energy of the largest superradiant state
is indicated by an integer, whereonemeans that the superradiant
state is in the lowest exciton state, two that it is in the next
excited state, etc. Asone can see for allmicrotubule segments with
number of spirals greater than 12, the superradiant state is in
thelowest exciton state. The large decaywidth of the superradiant
lowest exciton state indicates that such structurescould be able to
absorb photons ultra-efficiently. Indeed, the decaywidth of the
lowest exciton state of themicrotubule is in this case almost 600
times larger than the single-molecule decaywidth, corresponding to
avalue of roughly 1.64 cm−1. This translates to an absorbing time
scale of ÿ/Γ≈3.2 ps, which is very fast andcomparable with the
typical thermal relaxation times for biological structures [48],
suggesting that non-equilibriumprocessesmight be relevant in this
regime.
The superradiant state (that with the highest decaywidth) exists
for allmicrotubule segments, but thesuperradiant state coincides
with the lowest exciton state only for segments of 13 ormore
spirals. An intuitiveapproachwould suggest that, for 13 spirals,
themicrotubule topology becomes a ‘square’manifold:
13protofilaments×13 spirals, such that if the cylinder were cut
along its seam, onewould obtain a sheet oftubulins 13×13 square. At
this point the length of the cylinder (∼100nm) begins to far
outstrip itscircumference (∼75nm), and it is around this length
that supertransfer processes (see section 3.1, below)become
important, leading to the formation of a superradiant lowest
exciton state.
The existence of a superradiant lowest exciton state is
surprising considering that the positions andorientations of the
dipolesmay look quite disordered atfirst sight, as shown infigure
1(b). It is well known thatinteractions betweenmolecules can
destroy superradiance [36] unless dipole orientations have a
certain degreeof symmetry. The orientations of the Trp dipoles in
themicrotubule are far frombeing random, and theirsymmetry plays an
important role. To show this we consider two additionalmodels where
the positions of thedipoles are the same as in the realistic case
butwith their orientations randomized. First we consider the
casewhere the orientations of the dipoles are fully randomized over
thewholemicrotubule length. In such a case thesuperradiance is
completely suppressed, as shown infigure 2.Note also that infigure
2 the decaywidths aredistributed overmany states, in contrast to
the case of the native orientations of the dipoles, wheremost of
thedecaywidth of the system is concentrated in the lowest exciton
state. Themaximumvalue of the decaywidth forrandomized dipoles in
100 spirals ismuch smaller than that of the superradiant lowest
exciton state shown infigure 2, and even smaller than some
decaywidths shown infigure 2 for one spiral. Onemight also think
that theemergence of a superradiant lowest exciton state is
connectedwith the fact that the same dipole geometry isrepeated
over all the spirals. To show that this is not the case, we
considered a second randommodel withrandomorientations of dipoles
on a single spiral repeated over all other spirals. For this
partial randommodelwe still do not achieve a superradiant lowest
exciton state, as shown infigure 2.
In order to understand how the superradiant decaywidth increases
with the system size, infigure 3 themaximumdecaywidth is plotted as
a function of the length of themicrotubule segment. Note that the
decaywidth increases with the system size, but saturation occurs
when the length of themicrotubule is larger thanλ,thewavelength
associatedwith absorption by the transient dipoles.
Such a large decaywidth of the superradiant lowest exciton state
indicates that the excitation in the lowest-energy state is
extended overmanyTrpmolecules. In the upper panels offigure 4, the
probability offinding theexcitation on eachTrpmolecule (see
equation (6)) is shownwhen the systemof 100 spirals is in its
lowest-energy
Figure 3.Themaximumnormalized decaywidth of themicrotubuleΓmax/γ
is shown as a function of themicrotubule length Lrescaled by
thewavelength of the light that excites the atoms,λ=280 nm.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
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state. One can see that this state represents a fully extended
excitonic state over thewholemicrotubule segment,and thus it could
be capable of supporting ultra-efficient transport of
photoexcitation. Note that thesuperradiant lowest exciton state is
the statewhich ismost strongly coupled to the electromagnetic field
(highestdecaywidth), and thus the fact that it represents an
extended state implies that the absorbed photonwill beshared bymany
tryptophanmolecules in a coherent way, at least up to the dephasing
time. In the lower panels offigure 4, we also show for comparison
themost subradiant state for amicrotubule of 100 spirals. Note that
in thiscase the excitation probability is concentrated on the
chromophores of the innerwall of themicrotubule lumen,contrary to
the superradiant state where the excitation probability is
delocalized on the chromophores of theexternal wall that forms an
interface with the cytoplasm.
3.1. Structure of the superradiant lowest exciton state, super
and subtransfer processesIn order to understand the structure of
the superradiant lowest exciton state of a largemicrotubule
segment, wenowproject the lowest exciton state of thewhole
structure gsy ñ∣ not on the site basis as we did infigure 4,
butinstead onto alternative basis states: a basis mf ñ∣ made of the
eigenstates of a group of 13 coupled spirals. The ideais to take
amicrotubule segmentwhichwe can divide inmultiples of 13 spirals
and analyze which eigenstates of ablock of 13 spirals contribute to
form the superradiant lowest exciton state of thewholemicrotubule.
Note that13 is theminimumnumber of spirals we need to have a
superradiant lowest exciton state (see figure 2), and eachblock
ismade of nB=104×13=1352 states. If we call q
sy ñ∣ the eigenstate q of the s block of 13 spirals, then
thebasis state m q
sy yñ = ñ∣ ∣ for s n m sn1 B B- or m s n1 B -( ) . In the
upperpanel offigure 5, the first 13×104=1352 states correspond to
the eigenstates of thefirst 13 coupled spirals,the second 13×104
states correspond to the eigenstates of the coupled spirals from14
to 26, and so on. As onecan see from figure 5, the components of
the lowest exciton state over the 13 coupled spirals eigenstates
aremainly concentrated in the lowest exciton states of each block
of 13 coupled spirals (see also inset offigure 5upper panel). The
result infigure 5 clearly shows that the lowest exciton state of
thewhole structuremainlyconsists of a superposition of lowest
exciton states of smaller blocks of spirals. This non-trivial
result arises fromthe symmetry of the systems, see discussion in
[37]. A very interesting consequence of this is that the total
lowestexciton state emerges from coupling between the lowest
exciton states of smaller blocks. Such coupling is of
asupertransfer kind aswe showbelow.
The supertransfer coupling [4] between the lowest exciton states
of smaller blocks originates from theinteraction of the giant
dipolemoments associatedwith the superradiant lowest exciton states
of each block of 13spirals. Indeed, the lowest exciton state of a
block of 13 coupled spirals has a decay ratewhich is∼235 times
larger
Figure 4. Superradiant and subradiant excitonic states in
amicrotubule.The probability P x y z, , 2y=( ) ∣ ∣ of finding the
exciton on atryptophan chromophore of amicrotubule segment of 100
spirals with 10 400 tryptophanmolecules, is shown for the
extendedsuperradiant lowest exciton state (upper panels, lateral
view (left) and in cross section (right)) and themost subradiant
state (lowerpanels, lateral view (left) and in cross section
(right)), which has the smallest decay width (Γ/γ;10−8) and an
energy in themiddle ofthe spectrum (E e 2.1 cm0 1- = - ). Lengths
on each axis are expressed in nanometers. The participation ratio
from equation (12) forthe superradiant state in the upper panels is
3 512.92, and the participation ratio for themost subradiant state
in the lower panels is766.64.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
than the single-molecule decay rate. In order to prove the
previous statement, let us compute the couplingbetween the
eigenstates of two blocks of 13 coupled spirals, say block 1 and
block 2.Wewill compute thecoupling as a function of the distance
between the two blocks, assuming the blocks are translated along
theprincipal cylinder axis. Let us indicate the two corresponding
qth eigenstates of the two blocks as
C k ,s q
kks q, ,åy ñ = ñ∣ ∣
where the states kñ∣ represent the site basis of a block and
s=1, 2. The coupling between two single blockeigenstates can
bewritten as
V V C C V . 8q q q
k kk
qk
qk k12
1, 2,
,
1, 2,,*åy y= á ñ =
¢¢ ¢∣ ∣ ( ) ( )
Note that q1,yá ∣ is not the complex conjugate of q2,y ñ∣ but
the transpose of it, as we explain in section 2.Usingequations (1),
(4), and (5), we have thatV f r g ri 2k k k k k k k k k k k k, , ,
, ,m m= D - G = +¢ ¢ ¢ ¢ ¢ ¢
( ) · ( )r rk k k k k k, ,m m¢ ¢ ¢
( · ˆ )( · ˆ ), where the functions f, g can be derived from
equations (1), (4), and (5).When the distancebetween two blocks
ismuch larger than their diameter we can approximate r Rk k, 12Ȣ
whereR12 is the distancebetween the centers of the two blocks.
Equation (8) then becomes
V C C f R g R R R , 9q
k kk
qk
qk k k k12
,
1, 2,12 12 12 12*å m m m m= +
¢¢ ¢ ¢
( ) [ ( ) · ( )( · ˆ )( · ˆ )] ( )
where km
is the dipolemoment of the kmolecule. The above expression can
be re-written in terms of the dipole
strength of the eigenstates. The transition dipolemoment
Dqassociatedwith the qth eigenstate can be defined as
follows:
D C . 10qi
N
q i i1
,å m==
( )
The dipole coupling strength (often referred to as simply the
dipole strength) of the qth eigenstate is defined byDq 2
∣ ∣ (note that due to normalization D NnN
q12å ==
∣ ∣ ). Under the approximation that the imaginary part of
the
Hamiltonian (1) can be treated as a perturbation and L/λ=1we
have Dq q2 g» G
∣ ∣ (see appendix B). Thus,using equations (9), (10) can be
re-written as
V f R D g R D R D R . 11q q q q12 122
12 12 12*= +
[ ( )∣ ∣ ( )( · ˆ )( · ˆ )] ( )
Figure 5.Upper panel: projection of the lowest exciton state gsy
ñ∣ of amicrotubule segment of 91 spirals over the basis states mf
ñ∣ builtfrom the eigenstates of blocks of 13 spirals within the
segment (see text). The basis states, indexed bym, are ordered from
low to highenergy in each block (13 × 104=1352 states). The
projection has been computed as P mgs m gs m m gs
2 2f y f y= á ñ å á ñ( ) ∣ ∣ ∣ ∣ ∣ ∣ . The inset,zooming in on
thefirst nine eigenstates of thefirst block, confirms that the
lowest exciton state of thewholemicrotubule segment canbe viewed as
a coherent superposition of the lowest exciton states of the
smaller blocks of spirals. Lower panel: coupling between
thesuperradiant lowest exciton states of two blocks of 13 spirals
(blue circles) is comparedwith the average pairwise coupling
between thechromophores of each block (red squares) and themost
subradiant states of the two blocks (green triangles). The
couplings are plottedversus the distance d/λ (normalized by the
excitationwavelengthλ=280nm) between the centers of the two
blocks.When twoblocks are immediate neighbors, the center-to-center
distance is d≈116 nm. The supertransfer interaction between the
giant dipolesof the lowest exciton states of the two blocks (see
equation (11)), valid for large inter-block distances d, is shown
as a blue dashed curve.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
As a result for the coupling between the lowest exciton states
of blocks of 13 spirals, we obtainV D 235gs gs gs12
2 gµ » G »∣ ∣ ,Γgs/γ is the decay with of the lowest exciton
state of 13 spirals (note that we canuse the Dgs gs2 g» G∣ ∣
approximation since for a block of 13 spirals we have L/λ≈0.4). The
above expressionrepresents the interaction between the giant
dipoles of the lowest exciton states of each block. Therefore,
stateswith a large dipole strengthwill have a supertransfer
coupling proportional to the dipole strength of theeigenstates.
Note that the coupling between eigenstates with a small dipole
strength can give rise to asubtransfer effect, which has been shown
in [37]. In the lower panel of figure 5, the coupling between
thelowest exciton states with a large dipole strength (blue
circles) and between themost excited states with a verysmall dipole
strength (green triangles) of two blocks of 13 spirals is
comparedwith the average couplingbetween themolecules of each block
(red squares). Note that the lowest exciton state of a block of 13
spirals isthemost superradiant state withΓ/γ≈235, while the
highest-energy state is themost subradiant with thelowest
decaywidthΓ/γ≈10−6 for a block of 13 spirals. The couplings are
shown as a function of the center-to-center distance between the
two blocks normalized by the wavelength connected with the
opticaltransition. One can see that the coupling between the lowest
exciton states is significantly larger than theaverage coupling
between themolecules.Moreover, for large center-to-center distances
d, the couplingbetween the lowest exciton states is
well-approximated by equation (11) (see blue dashed curve), thus
provingthe existence of a supertransfer effect. On the other hand,
the coupling between themost excited states of thetwo blocks ismuch
smaller than the average coupling between themolecules, showing a
subtransfer effect.The above results suggest that the dynamics will
be very dependent on the initial conditions andwill exhibit atleast
two distinct timescales due to the presence of supertransfer and
subtransfer processes. In the nextsection, wewill showhow the
cooperativity-enhanced coupling between the lowest exciton states
of blocks ofspirals can boost photoexcitation transport.
4. Transport of photoexcitations via supertransfer
In this sectionwe consider the spreading velocity of an initial
excitation concentrated in themiddle of amicrotubulemade of 99
spirals, with a total length of about 800nm. The spreading of the
initial excitation hasbeenmeasured by the root-mean-square
deviation (RMSD) of the excitation along the longitudinal axis of
themicrotubule. Given that the initial state of the system is
described by thewavefunction 0y ñ∣ ( ) , the averageposition of the
excitation on the axis of themicrotubule can be computedwith Q t k
t zk k
2y= å á ñ( ) ∣ ∣ ( ) ∣ , wherezk is the position of the
kthmolecule on the longitudinal axis and ty ñ∣ ( ) is
thewavefunction at time t. Thus thevariance as a function of time
can be computedwith t k t z Q tk k
2 2 2 2s y= å á ñ -( ) ∣ ∣ ( ) ∣ ( ) , fromwhichfollows t tRMSD
2s=( ) ( ) .
We have chosen different initial conditions to show the effect
of cooperativity on the spreading of theexcitation: (i) an initial
excitation concentrated on a single randomly selected site of the
central spiral; and (ii) aninitial excitation concentrated in the
lowest exciton state of the central block of 5, 13, or 21 spirals.
As displayedinfigure 6, the spreading of the initial wave packet is
always ballistic ( t tRMSD µ( ) ) so that we can define avelocity
of spreadingV as the linear slope. Note thatV increases as we
increase the number of spirals over whichthe initial excitation is
spread and then saturates when the number of spirals becomes large.
Indeed from figure 6we can see that the spreading velocity is the
samewhen the initial state coincides with the lowest exciton state
of13 or 21 central spirals.When the excitation starts from the
lowest exciton state of 21 central spirals,V ismorethanfive times
the velocity of an excitation concentrated on a single site. Such
an effect is due to supertransfercoupling between the lowest
exciton states of blocks of spirals, and as a consequence of the
fact that the lowestexciton states of the central spirals have a
large overlapwith the extended superradiant state of
thewholemicrotubule, as shown infigure 5.
For comparisonwe also estimated the spreading velocity of an
excitationwhich can be expected basedon the typical
nearest-neighbor coupling present in the system. In the Trp case
the typical nearest-neighborcoupling is J≈50 cm−1, so that the time
needed for the excitation tomove by one Trp can be estimated as
cJ1 4 0.053t p= »( ) ps, where the light velocity is c≈0.03 cm
ps−1. This can be derived from the period ofoscillation between two
sites at resonance, which is given byT J hc cJ2 p= ¢ = ¢( ), and
τ=T/2 (note thatJ J hc= ¢ ( ) is the coupling in cm−1 asmeasured in
this paper). The average distance between Trps projectedalong
themain axis of themicrotubule can be evaluated from d=810 nm/10
400=0.078 nm.We thus obtaina velocityVNN=d/τ≈1.47 nm/ps, which is
shownby the black dashed line infigure 6.Note thatVNN is abouttwo
times smaller than the spreading velocity starting froma single
site. This is probably due to the fact that thelong-range
interactions between the sites favor the spreading of the
excitation (see section 5).Most importantly,VNN is about ten times
smaller then the spreading velocity of a delocalized excitation
obtained by setting theinitial state equal to the lowest exciton
state of 13 ormore central spirals. This strong dependence on
initial
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
conditionsmight also explain the variety of experimental data
obtained for diffusive excitonic behavior inmolecular
nanotubes.
For large times, the RMSD reaches a stationary value that
assumes the excitation becomes equally distributedon all Trps of
themicrotubule.We can compute such a stationary value of the RMSD
from the positions of the
Trps, by setting Q z Nk k= å and z N Qk k2 2 2s = å - , so that
RMSD 2s= . For amicrotubulemade of
99 spirals we obtain RMSD≈ 228 nm. This value is shown infigure
6 as a horizontal red line, and it agrees verywell with the
numerical results.
In realmicrotubules it can be the case that, due to various
decorations on the structure of the lattice(including other
interacting proteins and nucleotide di- and tri-phosphates, for
example), a localized excitationcould occur. In addition, due to
the presence of ultraweak light-emittingmolecules called reactive
oxygenspecies—the products of aerobic respiration in all biological
cells—such excitations can indeed be localized onindividual
chromophores of themicrotubule due to proximity. A fuller
description of these light-emittingmolecules and their impact on
tubulin polymers is given in [26].Wewould also like to emphasize
that a photonis likely to be absorbed by the superradiant state of
a block of spirals, since the superradiant state is the statewhich
ismost strongly coupled to the electromagnetic field (i.e. it has
the highest decaywidth and absorptionrate). For this reason an
initial excitation coinciding with the superradiant state of a
block ofmicrotubule spiralsis wellmotivated, and the fact that its
spreading is enhanced can have important consequences
forphotoexcitation transport.
Finally, in order to emphasize the role of symmetry in the
transport properties of the system, in the rightpanel offigure 6we
show the spreading of an initial excitation starting froma single
site on the central spiral forthe realisticmodel considered before
(black circles in both left and right panels represent the same
data), for thefully randommodel, and for the partial randommodel.
In both the lattermodels, the positions of the dipoles
arekeptfixedwith respect to the realisticmodel, but the orientation
of their dipoles has been randomized. Note thatin the fully
randommodel the dipole directions have been randomized along the
entiremicrotubule length,whereas for the partial randommodel we
have randomized the dipole orientations in one spiral and
thenrepeated this configuration in all other spirals. For the
partial randommodel, the excitation spreads over
thewholemicrotubule segment with a smaller velocity than the
realisticmodel, while for the fully randommodelthe spreading is
extremely slow and remainswell below the value (see horizontal red
line) of an equallydistributed excitation during thewhole
simulation time. The above results show the relevance of
nativesymmetry in excitonic energy transport through
themicrotubule.Wewould like to point out that several recentworks
have highlighted the role of symmetry in enhancing coherent [37]
and transport properties ofmolecularnetworks [49].
Figure 6.Root-mean-square deviation (RMSD) as a function of time
for an initial excitation concentrated in themiddle of amicrotubule
comprised of 99 spirals with a total length of about 810nm. Left
panel: the case of an initial excitation concentrated on asingle
site of the central spiral (black circles) is comparedwith an
initial condition on the lowest exciton state of the central five
(emptygreen squares), 13 (orange crosses), and 21 (blue stars)
spirals. For comparison, the spreading expected from the strength
of thenearest-neighbor coupling is also shown (black dashed line)
and described in the text. The equilibrium value for an excitation
equallydistributed over all the tryptophans of themicrotubule is
shown as a red horizontal line. The spreading velocity of an
excitationstarting from a single site is about two times faster
than the spreading associatedwith the amplitude of the
nearest-neighbor coupling(NNvelocity), while the spreading velocity
of an excitationwhich starts in the lowest exciton state ofmore
than 13 spirals is about tentimes faster than theNNvelocity. Right
panel: here the case of an initial excitation concentrated on a
single site of the central spiral isconsidered for three
differentmodels—the realisticmodel (black circles and same data as
in the left panel), the fully randommodelwhere all the dipoles are
randomly oriented (filled green squares), and the partial
randommodel where the dipoles are oriented atrandombut repeated in
the same configuration for all spirals (blue crosses).
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
5. Robustness to disorder and the role of long-range
interactions
In order to study the robustness to disorder of the superradiant
lowest exciton state, we have analyzed the systemin the presence of
static disorder, i.e. time-independent and space-dependent
fluctuations of the excitationenergies of the tryptophans
comprising themicrotubule chromophoric lattice. Specifically we
consider that theexcitation energies of the tryptophans are
uniformly distributed around the initial value e0, between e W 20
-and e W 20 + , so thatW represents the strength of the static
disorder. It is well known that static disorderinduces localization
of the eigenstates of a system, a phenomenon known as Anderson
localization [50]. Due tosuch localization, for each eigenstate the
probability offinding the excitation is concentrated on very few
sites forlarge disorder, and only on one site for extremely large
disorder. Note that Anderson localization usually occursin the
presence of short-range interactions, but in ourmodel there
aremultiple contributions from acomplicated power law for the
interaction (see equation (4)), so that the results of our analysis
are not obvious.
In order to study the robustness of superradiance to disorder,
we plot in figure 7 themaximumnormalizeddecaywidthΓSR/γ as a
function of the disorder strengthW, using the full realistic
non-HermitianHamiltoniangiven in equation (1) for
differentmicrotubule sizes. Note that themost superradiant state
(i.e. largest decaywidth) coincides with the lowest exciton state
of themicrotubule for sizes of 13 ormore spirals. One can see
thatthe disorder at which thewidth of the superradiant state starts
to decrease is independent of the system size(within the system
sizes considered in our simulations). This is quite surprising for
quasi-one dimensionalstructures, which usually exhibit a critical
disorder that decreases as the system size grows [51]. Indeed, for
short-range interactions in quasi-one dimensional structures, the
critical static disorder strengthWcrneeded tolocalize the system
goes to zero as the system size goes to infinity, withW J Ncr µ
.
In order to understand how the above results could be explained
by the effective range of the interaction, wehave compared our
realisticmodel which contains long-range interactions with the
samemodel where the long-range interactions have been suppressed.
Specifically the short-rangemodel has been obtained by
consideringonly the interactions between Trpswith a
center-to-center separation less than 4nm. In order to perform such
acomparison, only theHermitian part of the
realisticmodelHamiltonian given in equation (1) has been takeninto
account.We considered only theHermitian part of theHamiltonian
because it constitutes a goodapproximation of thewholeHamiltonian
(see discussion in appendix B) and,most importantly, allows
forcomparison of different ranges of interaction. Indeed, in the
full non-Hermitianmodel we cannot change therange of the
interactionwithout introducing inconsistencies (i.e. negative
decaywidths). Belowwewill show thatthe results thus obtained are
consistent with the analysis of thewholeHamiltonian given infigure
7, where thefull non-HermitianHamiltonian has been considered.
We analyzed the effect of disorder for the two differentmodels
(long-range and short-range) through theparticipation ratio
(PR)
q qPR 12i
ii
i2 4å åy y= á ñ á ñ∣ ∣ ∣ ∣ ∣ ∣ ( )
of the eigenstates yñ∣ of the system,where the large outer
brackets, ...á ñ, stand for the ensemble average overdifferent
realizations of the static disorder. The PR is widely used to
characterize localization properties [52], and
Figure 7.Normalized decay width of the superradiant state is
plotted versus the strength of static disorderW for
amicrotubulesegment of different lengths.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
it satisfies the bounds N1 PR . For extended states, it
increases proportionally to the system sizeN, while,for localized
states, it is independent ofN.
In order to study the effect of static disorder, we have
analyzed the PR of the lowest exciton state as a functionof the
disorder strengthW infigure 8. As shown in the upper panel offigure
8, where the long-rangemodel isanalyzed, the critical disorder at
which the PR starts to decrease appears to be independent of
themicrotubulelength. The critical disorder obtained in this case
is consistent with the analysis of the critical disorder needed
toquench superradiance as shown in figure 7. The response of the
system to disorder is completely different for theshort-rangemodel.
The robustness to disorder of the lowest exciton state of such
amodel is shown in the lowerpanel offigure 8. As one can see, the
critical disorder decreases with the system size in this case, as
would beexpected for quasi-one dimensional systemswith short-range
interactions. The difference between the twopanels offigure 8 shows
that the long-range nature and symmetry of the interactions play a
very significant rolein enhancing the robustness of excitonic
states inmicrotubules to disorder. However, we note that robustness
todisorder could also be connected to supertransfer, and not only
to the long-range of the interaction. For furtherdetails see the
discussion in [37] and in appendix C.
6. Conclusions and perspectives
Wehave analyzed the excitonic response ofmicrotubules induced by
the coupling of tryptophanmolecules,which are themost strongly
photoactivemolecules in the spiral-cylindrical lattice. The
positions andorientations of the dipoles of the tryptophanmolecules
have been obtained in previous works bymoleculardynamics
simulations and quantum chemistry calculations and have closely
reproduced experimental spectrafor the tubulin heterodimeric
protein [26]. Analyzing the properties of amicrotubule of length
L∼800 nm,which is larger than thewavelength of the excitation
transition (L>λ=280nm), requires an approach thatgoes beyond the
transition dipole–dipole couplings alone. This is whywe take into
consideration radiativeinteractions containing non-Hermitian
terms.
Our analysis has shown that the coupling between
tryotophanmolecules is able to create a superradiantlowest exciton
state, similar to the physical behavior of several photosynthetic
antenna systems. Such asuperradiant lowest exciton state, which
absorbs in theUV spectral range, has been shown to be a
coherentexcitonic state extended over thewholemicrotubule lattice
of tryptophanmolecules. Interestingly, thesuperradiant lowest
exciton state appears to be delocalized on the exterior wall of
themicrotubule, whichinterfaces with the cytoplasm, suggesting the
possibility that these extended but short-lived (few
picosecond)excitonic statesmay be involved in communicationwith
cellular proteins that bind tomicrotubules in order to
Figure 8.Participation ratio (PR) of the lowest exciton state
for amicrotubule of different lengths is shown versus the strength
of staticdisorderW, averaged over ten realizations of disorder for
each system length. The upper panel presents the results of the
fullmodel,where the interaction between the tryptophans has a
long-range nature (see equation (1)). However, note that only
theHermitian partof theHamiltonian in equation (1)has been used to
obtain the data shown in both panels of thisfigure. In the lower
panel, the PRofthe lowest exciton state for amicrotubule of
different lengths is shown versus the strength of static disorderW
for the case of short-range interactions only, considering only the
couplings between tryptophanswith a center-to-center separation
smaller than 4nm.
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New J. Phys. 21 (2019) 023005 GLCelardo et al
-
carry out their functions. At the same time, we have shown that
long-lived (hundreds ofmilliseconds)subradiant states can be
concentrated on the inner wall of themicrotubule lumen. These
subradiant states couldbe particularly important in the
synchronization of neuronal processes in the brain,
wheremicrotubules canextend to themicron scale and beyond.
Moreover, we have shown that the superradiant lowest exciton
state of thewholemicrotubule arises througha supertransfer coupling
between the lowest exciton states of smaller blocks of spirals
within themicrotubule.For this reason,microtubule superradiance is
essentially an emergent property of thewhole system that developsas
‘giant dipole’ strengths of superradiant lowest exciton states of
constituent blocks interact to form adelocalized coherent state on
the entire structure. This is a hallmark of self-similar behavior,
in the sense thatsubunit blocks of spirals exhibit superradiant
characteristics that recapitulate roughlywhat is seen in
thewhole.Only by considering the entire structure (or at least a
substantial fraction of the spirals) dowe uncovercooperative and
dynamical features of the system that would otherwise fail to be
captured inmore reductionistmodels. Supertransfer coupling between
excitonic states of different blocks of spirals in
themicrotubulesegment is critical to themanifestation of these
cooperative behaviors and explains the calculated couplings
toexcellent agreement.
Such supertransfer coupling is able to enhance the spreading of
photoexcitation inside themicrotubule. Thespreading of
photoexcitation is ballistic, despite the fact that the native
dipole orientations of the tryptophanmolecules are not fully
symmetric even in the absence of static disorder (see figure 1).
The spreading velocity isstrongly dependent on the initial
conditions, and, due to supertransfer, it can be about ten times
faster than thevelocity expected from the amplitude of the
nearest-neighbor coupling between the tryptophanmolecules insuch
structures. These results show that the characteristic
supertransfer processes analyzed in photosyntheticantenna
complexesmay also be present inmicrotubules.
Finally, we have analyzed the robustness ofmicrotubule
superradiance to static disorder.We have shownthat the symmetry and
long-range nature of the interactions give an enhanced robustness
to such structures witha critical disorder which appears to be
independent of the system size (up to the system sizes analyzed in
thispaper). This is at variancewithwhat usually happens in
quasi-one dimensional structures with short-rangeinteractions,
where the critical disorderWcr goes to zero as the system size
grows. Indeed, for quasi-onedimensional systemswith short-range
interactions only, we haveW J Ncr µ , where J is the
nearest-neighborcoupling andN is the number of chromophore sites.
The critical disorder at which superradiance and thedelocalization
of the lowest exciton state are precipitously affected is found to
be on the order of 10 cm−1 (seefigure 8). Such a value of disorder
is not extremely large, as natural disorder can be on the order of
kT, rangingfrom50 to 300 cm−1. Still, such critical disorder ismuch
larger than the critical disorder expected from thetypical
nearest-neighbor coupling between tryptophanmolecules. Such
enhanced robustness to static disorderas a result of long-range
interactions and symmetry can greatly increase diffusion lengths
and thereby supportultra-efficient photoexcitation transport.
To refine our studies, future work should certainly include
consideration of the other photoactive aminoacids, such as the
aromatics tyrosine and phenylalanine, that are present
inmicrotubules, as well as the effects ofthermal relaxation on
coherent energy transport. To realistically take into account such
environmental effects,wemust analyze the role of
out-of-equilibriumprocesses, which can dominate the dynamics of the
properfunctioning state of themicrotubule before thermalization.
Environmental decoherencewill be consideredexplicitly in a
futurework.Our contribution in this paper is to provide a first
analysis andmotivation for suchfurther nuanced studies of excitonic
states in this intriguing and highly symmetric biological
system.
The significance of photoexcitation transport inmicrotubules is
an open question in the biophysicscommunity, and further
experimental and theoretical works are needed to establish the
precisemechanisms oftheir optical functionality. Our results point
towards a possible role of superradiant and supertransfer
processesinmicrotubules. Both cooperative effects are able to
induce ultra-efficient photoexcitation absorption and couldserve to
enhance energy transport over long distances under natural
conditions.We hope that our results willinspire further
experimental studies onmicrotubules to gather evidence forUV
superradiance in suchimportant biological structures.
It would be interesting to further contextualize the biological
implications of our results. In a series of studiesspanning a
period of almost a quarter century, Albrecht-Buehler observed that
living cells possess a spatialorientationmechanism located in the
centrosome [35, 53–55]. The centrosome is formed from an
intricatearrangement ofmicrotubules in two perpendicular sets of
nine triplets (called centrioles). This arrangementprovides the
cell with a primitive ‘eye’ that allows it to locate the position
of other cells within a two-to-three-degree accuracy in the
azimuthal plane andwith respect to the axis perpendicular to it
[55]. Though it is still amystery how the reception of
electromagnetic radiation is accomplished by the centrosome,
superradiantbehavior in thesemicrotubule aggregatesmay play a
role.
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Acknowledgments
GLCacknowledges the support of PRODEP (511-6/17-8017). TJACwould
like to acknowledge financialsupport from theDepartment of
Psychology andNeuroscience and the Institute
forNeuro-ImmuneMedicineatNova SoutheasternUniversity (NSU), andwork
in conjunctionwith theNSUPresident’s Faculty
ResearchandDevelopmentGrant (PFRDG) programunder PFRDG335426
(Craddock PI). PKwas supported in part bytheNational Institute
onMinorityHealth andHealthDisparities of theNational Institutes
ofHealth underAwardNumberG12MD007597. The content is solely the
responsibility of the authors and does not necessarilyrepresent the
official views of theNational Institutes ofHealth. PKwould also
like to acknowledge support fromtheUS-Italy Fulbright Commission
and theWholeGenome Science Foundation.
AppendixA.Microtubule tryptophan dipole positions and
orientations
The tubulinα–β heterodimer structurewas obtained by repairing
the protein data bank (PDB:www.rcsb.org)[56] structure 1JFF [57] by
addingmissing residues from1TUB [58] after aligning 1TUB to 1JFF.
This initialrepaired dimer structurewas oriented by itself alone
such that the (would be) protofilament direction alignedwith the
x-axis, the normal to the (would be) outermicrotubule surface
alignedwith the y-axis, and the directionof (would be) lateral
contacts alignedwith the z-axis, before subsequent translation and
rotation. A single spiralof 13 tubulin dimerswas generated by
translating each dimer 11.2 nanometers (nm) in the y-direction,
thensuccessively rotating the resulting dimer structure bymultiples
of −27.69° in the y–z plane about the originaround the x-axis, and
successively shifting each dimer bymultiples of 0.9 nanometers in
the x-direction. Thisresulted in a left-handed helical-spiral
structurewith a circular radius of 22.4 nmpassing through the
center ofeach dimer in the B-latticemicrotubule geometry described
by Li et al [59] and Sept et al [60]. The orientation ofthe 1La
excited state of each tryptophanmolecule in the resulting
structurewas taken as 46.2° above the axisjoining themidpoint
between theCD2 andCE2 carbons of tryptophan and carbonCD1, in the
plane of theindole ring (i.e. towards nitrogenNE1). TheCartesian
positions and unit vector directions of the 104tryptophans of
thefirst spiral are given in table A1 below. To generate successive
spirals, the initial spiralcoordinates were translated along the x
(i.e. protofilament) direction bymultiples of 8 nm.Modelingwas
donewith PyMOL1.8.6.2 [61].
TableA1. First spiral dipole positions (Å) and unit vectors.
x y z xm̂ ym̂ zm̂
−2.378 103.218 14.720 −0.701 14 0.665 10 −0.256 99−13.691
123.899 7.109 0.654 56 −0.702 98 −0.278 1613.384 124.916 −9.487
−0.538 55 −0.005 10 −0.842 58−28.566 122.415 5.686 −0.185 73 -0.217
51 −0.958 226.622 97.563 −31.783 −0.701 25 0.469 22 −0.536 74−4.691
112.339 −48.133 0.654 55 −0.751 73 0.080 3922.384 105.527 −63.300
−0.538 54 −0.395 97 −0.743 76−19.566 110.363 −48.703 −0.185 68
−0.638 15 −0.747 1815.622 70.946 −70.331 −0.701 34 0.165 74 −0.693
294.309 76.431 −91.675 0.654 56 −0.628 18 0.420 6431.384 63.350
−101.939 −0.538 42 −0.696 43 −0.474 43−10.566 74.417 −91.261 −0.185
63 −0.912 28 −0.365 0924.622 29.463 −92.094 −0.701 42 −0.175 12
−0.690 9013.309 24.401 −113.542 0.654 55 −0.360 84 0.664 3540.384
8.049 −116.552 −0.538 45 −0.837 15 −0.096 15−1.566 22.810 −112.239
−0.185 84 −0.977 46 0.100 2233.622 −17.382 −92.086 −0.701 21 −0.476
30 −0.530 5222.309 −31.831 −108.725 0.654 55 −0.010 98 0.755
9449.384 −47.710 −103.791 −0.538 40 −0.786 05 0.303 747.435 −32.636
−106.832 −0.185 61 −0.818 78 0.543 2742.622 −58.858 −70.309 −0.701
12 −0.668 44 −0.248 2531.309 −79.384 −78.327 0.654 55 0.341 93
0.674 2758.384 −91.151 −66.579 −0.538 40 −0.554 61 0.634 4516.435
−79.217 −76.278 −0.185 81 −0.472 52 0.861 5151.622 −85.462 −31.752
−0.701 43 −0.706 94 0.090 6740.309 −107.363 −29.312 0.654 56 0.616
02 0.438 2567.384 −112.323 −13.442 −0.538 47 −0.196 91 0.819
3225.435 −106.263 −27.576 −0.185 98 −0.018 12 0.982 39
15
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http://www.rcsb.org
-
TableA1. (Continued.)
x y z xm̂ ym̂ zm̂
60.622 −91.101 14.753 −0.701 19 −0.584 26 0.408 6349.309
−109.360 27.091 0.654 54 0.749 18 0.101 5376.384 −106.376 43.448
−0.538 46 0.206 79 0.816 8834.435 −107.578 28.118 −0.185 79 0.440
42 0.878 3669.622 −74.482 58.551 −0.701 22 −0.327 15 0.633 4658.309
−84.916 77.961 0.654 57 0.710 67 −0.257 8485.384 −74.672 91.058
−0.538 50 0.562 68 0.627 2343.435 −82.861 78.042 −0.185 66 0.798 24
0.573 0178.622 −39.413 89.609 −0.701 01 0.004 58 0.713 1367.309
−39.631 111.644 0.654 60 0.509 89 −0.558 1494.384 −24.474 118.481
−0.538 46 0.789 61 0.294 2452.435 −37.774 110.761 −0.185 83 0.973
05 0.136 5387.622 6.073 100.812 −0.701 34 0.335 63 0.628 8776.309
16.121 120.425 0.654 58 0.191 39 −0.731 37103.384 32.718 119.434
−0.538 49 0.835 84 −0.106 7861.435 17.355 118.780 −0.185 56 0.924
97 −0.331 6896.622 51.555 89.594 −0.701 37 0.589 28 0.401 0485.309
69.566 102.291 0.654 56 −0.170 71 −0.736 49112.384 83.802 93.700
−0.538 47 0.690 44 −0.483 0570.435 69.894 100.261 −0.185 84 0.665
06 −0.723 30105.622 86.614 58.524 −0.701 31 0.708 20 0.081 3894.309
108.463 61.397 0.654 56 −0.493 12 −0.573 04121.384 117.076 47.174
−0.538 52 0.387 15 −0.748 4179.435 107.810 59.447 −0.185 62 0.252
46 −0.949 6439.057 102.384 14.619 −0.017 22 −0.525 00 0.850
9353.331 124.899 −8.386 −0.683 14 0.492 02 −0.539 6736.453 121.233
−4.011 0.988 86 0.147 94 0.016 6312.616 122.434 5.625 −0.190 86
−0.224 02 −0.955 7148.057 96.778 −31.485 −0.016 90 −0.069 07 0.997
4762.331 106.023 −62.318 −0.683 15 0.184 96 −0.706 4745.453 104.810
−56.739 0.988 82 0.138 86 −0.054 3721.616 110.352 −48.767 −0.190 62
−0.642 62 −0.742 1057.057 70.389 −69.702 −0.016 92 0.402 17 0.915
4171.331 64.247 −101.300 −0.683 15 −0.164 73 −0.711 4654.453 65.765
−95.796 0.988 82 0.097 92 −0.112 4930.616 74.376 −91.313 −0.190 47
−0.913 72 −0.358 9366.057 29.262 −91.278 −0.017 30 0.781 43 0.623
7680.331 9.139 −116.402 −0.683 14 −0.476 49 −0.553 4263.453 13.041
−112.235 0.988 85 0.034 27 −0.144 9439.616 22.750 −112.267 −0.190
77 −0.975 79 0.106 9875.057 −17.181 −91.270 −0.017 07 0.981 86
0.188 8389.331 −46.675 −104.165 −0.683 12 −0.679 18 −0.268 4572.453
−41.283 −102.288 0.988 85 −0.037 02 −0.144 2248.616 −32.701
−106.829 −0.190 59 −0.814 21 0.548 3984.057 −58.300 −69.680 −0.017
07 0.957 20 −0.288 9198.331 −90.408 −67.392 −0.683 09 −0.726 12
0.078 3481.453 −84.762 −68.236 0.988 89 −0.099 73 −0.110 2557.616
−79.273 −76.244 −0.190 47 −0.466 22 0.863 9293.057 −84.676 −31.454
−0.017 21 0.713 39 −0.700 55107.331 −112.043 −14.506 −0.683 10
−0.606 69 0.406 5890.453 −107.435 −17.878 0.988 81 −0.140 01 −0.051
6066.616 −106.297 −27.520 −0.190 64 −0.011 55 0.981 59102.057
−90.266 14.651 −0.017 06 0.305 98 −0.951 89116.331 −106.622 42.375
−0.683 16 −0.348 62 0.641 6899.453 −104.109 37.249 0.988 80 −0.148
00 0.019 3575.616 −107.582 28.183 −0.190 53 0.445 82 0.874
61111.057 −73.790 58.073 −0.017 08 −0.171 42 −0.985 05125.331
−75.389 90.223 −0.683 11 −0.009 97 0.730 25108.453 −75.546 84.516
0.988 84 −0.121 78 0.085 7784.616 −82.834 78.102 −0.190 55 0.801 45
0.566 89120.057 −39.023 88.865 −0.017 03 −0.609 39 −0.792 69134.331
−25.497 118.074 −0.683 14 0.330 05 0.651 45117.453 −28.288 113.094
0.988 88 −0.067 76 0.132 36
16
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-
Appendix B. Comparison between dipole–dipole and
radiativeHamiltonians
Herewe compare the dipole–dipoleHamiltonian (Dipolemodel), which
includes only theHermitian part of thecoupling in equation (7),
with the full radiative non-HermitianHamiltonian given in equations
(1)–(5) (nHmodel).Wewill also include in our analysis theHermitian
part of the full radiative non-HermitianHamiltonian(Hmodel). For
the threemodels (Dipole, nH, andH)we compare both the real-valued
energies and the dipolecoupling strengths of their
eigenstates.Wewill show that, in the small-volume limit L/λ=1, both
quantitiescan be computedwith the threemodels, but when the system
size is larger than thewavelength, only the nHmodel can be used to
compute the dipole strengths of the eigenstates. However, theHmodel
still gives a closeestimation to the nHmodel for the real energies
in the large-volume limit, though theDipolemodel displaysdeviations
from the nHmodel values.
Infigure B1, we compare the real part of the spectrum for the
threemodels, focusing on the eigenvaluesclose to the lowest exciton
state. In the upper panel, we present amicrotubulemade of only one
spiral, so thatL 1l . In this case the threemodels all give very
similar estimations of the eigenvalues. In the lower panel,the case
of amicrotubule of 100 spirals is considered. In this case the
system size is not small comparedwith thewavelength, as L/λ≈3.One
can see that while theHmodel is a very good approximation of the
nHmodel,theDipolemodel exhibitsmaximumdeviations of ∼1 cm−1 at and
near the lowest exciton state.
When the system size is small compared to thewavelength
associatedwith the optical transition of themolecules, the optical
absorption of an eigenstate of the aggregate can be estimated in
terms of its dipolestrength, computed only from theHermitian part
of theHamiltonian (1). Denoting the nth eigenstate of theHermitian
part of theHamiltonian (1) or of theHamiltonianwith onlyHermitian
coupling in (7) as Enñ∣ , we canexpand it in the site basis, so
that
E C i . B.1ni
N
ni1
åñ = ñ=
∣ ∣ ( )
Note that the site basis is referred to by the
tryptophanmolecules and is composed of the states iñ∣ , each of
themcarrying a dipolemoment im
. IfN is the total number ofmolecules, thenwewill express the
transition dipole
moment Dnassociatedwith the nth eigenstate as follows:
D C . B.2ni
N
ni i1
å m==
ˆ ( )
The dipole coupling strength (often referred to as simply the
dipole strength) of the nth eigenstate is defined byDn 2
∣ ∣ (note that due to normalization D NnN
n12å ==
∣ ∣ ). Under the approximation that L/λ= 1we have
Dn n2 g» G
∣ ∣ , whereΓn is given by the imaginary part of the complex
eigenvalues E i 2n n n = - G of the nHmodel. On the other hand, in
the large-volume limit, the dipole as defined above in equation
(B.2) gives incorrectresults and does not represent the dipole of
the eigenstates. This is shown infigure B2, where themaximumdipole
strength computed using theDipolemodel and theHmodel is
comparedwith themaximumdecaywidthΓmax/γ computedwith the full
radiative nHmodel. As one can see, the dipole coupling strength
computed asdescribed above is valid only for small system
sizes.
TableA1. (Continued.)
x y z xm̂ ym̂ zm̂
93.616 −37.723 110.802 −0.190 64 0.973 06 0.129 64129.057 6.073
99.971 −0.017 07 −0.908 05 −0.418 51143.331 31.624 119.550 −0.683
13 0.595 19 0.423 18126.453 26.838 116.437 0.988 91 0.001 53 0.148
53
102.616 17.419 118.792 −0.190 54 0.921 88 −0.337 38138.057
51.164 88.850 −0.017 32 −0.998 53 0.051 31152.331 82.887 94.311
−0.683 12 0.723 71 0.097 98135.453 77.203 93.780 0.988 88 0.070 31
0.131 06
111.616 69.956 100.242 −0.190 74 0.659 33 −0.727 26147.057
85.922 58.046 −0.017 08 −0.860 20 0.509 68161.331 116.549 48.140
−0.683 14 0.686 44 −0.249 26144.453 111.269 50.311 0.988 85 0.123
36 0.083 42
120.616 107.856 59.401 −0.190 53 0.245 75 −0.950 42
17
New J. Phys. 21 (2019) 023005 GLCelardo et al
-
AppendixC. Supertransfer and the energy gap in the complex
plane
Wewould like to point out that supertransfermight also play an
important role in stimulating robustness todisorder. For instance,
infigure C1we show the energy differences in the complex plane
between the lowestexciton state (which coincides with themost
superradiant state for amicrotubule of 13 ormore spirals) and
thenext excited state. As one can see, the energy gap increases
with the system size, instead of decreasing as onewould expect, for
lengths up to the excitationwavelength. Such counterintuitive
behavior for the energy gap hasanalogously been found in
photosynthetic complexes by two of the authors of this paper [37],
where it has beenconnected to the presence of supertransfer. It is
well known that such energy gaps can protect states fromdisorder,
but the precise consequences for robustness of this gap in
cylindrical aggregates need to be studiedmore carefully.We plan to
do this in the future.
Figure B1. Lowest part of the spectrum (real-valued energies En
versus eigenstate index n) for amicrotubule of 1 spiral,
L/λ=1,(upper panel) and 100 spirals, L/λ≈3, (lower panel) is shown
for the three differentmodels considered (seemain text). In the
small-volume limit (upper panel), all threemodels give similar
estimations of the spectrum, but in the large-volume limit (lower
panel) theDipolemodel deviates from theH and nHmodels, which are
very close to each other.
Figure B2.Maximumdipole coupling strength Dmax 2∣ ∣ computed
from equation (B.2) for theDipole andHmodels (seemain text)
iscomparedwith the relative decaywidthΓmax/γ computed from the full
radiative nHmodel (seemain text) as a function of the systemsize
Lnormalized by the excitationwavelength (λ=280nm).
18
New J. Phys. 21 (2019) 023005 GLCelardo et al
-
ORCID iDs
PKurian https://orcid.org/0000-0002-4160-6434
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1. Introduction2. The model3. Superradiance in the lowest
exciton state3. Superradiance in the lowest exciton state3.1.
Structure of the superradiant lowest exciton state, super and
subtransfer processes
4. Transport of photoexcitations via supertransfer5. Robustness
to disorder and the role of long-range interactions6. Conclusions
and perspectivesAcknowledgmentsAppendix A.Appendix B.Appendix
C.References